Galaxy mass models: MOND versus dark matter haloes

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1 MNRAS (2014) MNRAS Advance Access published February 17, 2014 doi: /mnras/stu100 Galaxy mass models: MOND versus dark matter haloes Toky H. Randriamampandry and Claude Carignan Department of Astronomy, University of Cape Town, Private Bag X3, Rondebosch 7701, South Africa Accepted 2014 January 14. Received 2013 December 19; in original form 2013 July 18 1 INTRODUCTION Since the 1970s, it is well known that there is a discrepancy between the visible mass and the dynamical mass of galaxies (e.g. Freeman 1970; Shostak 1973; Roberts & Whitehurst 1975; Bosma 1978). The commonly accepted explanation of the galaxy mass discrepancy is to assume a more or less spherical halo of unseen dark matter (DM) in addition to the visible baryonic mass in the form of stars and gas (see e.g. Carignan & Freeman 1985). The distribution of DM in galaxies can be defined by a theoretical or an empirical density distribution profile. Several density profile models for the distribution of DM are presented in the literature. The two most commonly used models are (see also e.g. Einasto 1969; Burkert 1995; Kravtsovetal.1998): (i) the pseudo-isothermal (ISO) DM halo model (observationally motivated model), (ii) the Navarro Frenk White (NFW) DM halo model (Navarro, Frenk & White 1997) (derived from CDM N-body simulations) The highest quality H I RCs available to date are those for the THINGS (The H I Nearby Galaxies Survey) sample (Walter et al. 2008). For that sample, the rotation curves (RCs) of 19 galaxies were derived by de Blok et al. (2008) and their mass distributions were modelled using the ISO and the NFW halo density profiles. de Blok tokyr@ast.uct.ac.za ABSTRACT Mass models of 15 nearby dwarf and spiral galaxies are presented. The galaxies are selected to be homogeneous in terms of the method used to determine their distances, the sampling of their rotation curves (RCs) and the mass-to-light ratio (M/L) of their stellar contributions, which will minimize the uncertainties on the mass model results. Those RCs are modelled using the MOdified Newtonian Dynamics (MOND) prescription and the observationally motivated pseudo-isothermal (ISO) dark matter (DM) halo density distribution. For the MOND models with fixed M/L, better fits are obtained when the constant a 0 is allowed to vary, giving a mean value of (1.13 ± 0.50) 10 8 cm s 2, compared to the standard value of cm s 2. Even with a 0 as a free parameter, MOND provides acceptable fits (reduced χr 2 < 2) for only 60 per cent (9/15) of the sample. The data suggest that galaxies with higher central surface brightnesses tend to favour higher values of the constant a 0. This poses a serious challenge to MOND since a 0 should be a universal constant. For the DM models, our results confirm that the DM halo surface density of ISO models is nearly constant at ρ 0 R C 120 M pc 2.This means that if the M/L is determined by stellar population models, ISO DM models are left with only one free parameter, the DM halo central surface density. Key words: galaxies: dwarf galaxies: kinematics and dynamics galaxies: spiral galaxies: structure dark matter. et al. (2008) found that most of the galaxies in their sample preferred the observationally motivated core-dominated ISO halo over the cuspy NFW halo. This is known as the core cusp controversy (see de Blok 2010 for a review). This is why the ISO models will be used in this paper for the DM models. An alternative to the missing mass problem is the MOdified Newtonian Dynamics (MOND; Milgrom 1983a,b,c). Milgrom postulates that at small accelerations the usual Newtonian dynamics break down and that the law of gravity needs to be modified. MOND claims to be able to explain the mass discrepancies in galaxies without the need for DM but with the introduction of a universal constant a 0, which has the dimension of an acceleration. The phenomenological success of MOND to reproduce the observed RCs of galaxies has attracted a huge interest in the astronomy community for the last three decades. One of the first studies was made by Kent (1987) usinghi RCs of spiral galaxies and his conclusions were not favourable to MOND. Kent s work was criticized by Milgrom (1988) who pointed out possible errors in the distances and inclinations used and the possibility that not all the H I gas had been detected. Dwarf galaxies are critical to test a theory such as MOND because these objects present the largest discrepancies between the visible mass and the dynamic mass and their small accelerations put them mostly into the MOND regime. Lake (1989) concluded that MOND could not reproduce the observed RCs of a sample of dwarf galaxies where the luminous mass was dominated by the gas and not the stars. However, Milgrom (1991) disapproved Lake s conclusions C 2014 The Authors Published by Oxford University Press on behalf of the Royal Astronomical Society

2 2 T. Randriamampandry and C. Carignan arguing again that there could be large errors in the distances and inclinations. Begeman, Broeils & Sanders (1991) were the first to estimate the value of a 0 by fitting the RCs of bright spiral galaxies. The MOND parameter a 0 was taken as a free parameter during their fitting procedure. They found an average value of a 0 = cm s 2. However, the distances used were quite uncertain since Hubble s Law was adopted as the distance indicator for most of the galaxies in their sample. The Begeman et al. (1991) results were confirmed by Sanders (1996) and Sanders & Verheijen (1998) using the same method. Sanders (1996) used a larger sample of 22 galaxies selected from the literature but only M33 and NGC 300 had Cepheid distances. Since then, the value found by Begeman et al. (1991) is considered as the standard value for a 0. Bottema et al. (2002) were the first to use a sample with Cepheid-based distances only and found that a lower value for a 0 ( cm s 2 ) yields better fits to the RCs. A complete review about the previous tests of MOND is presented in Sanders & McGaugh (2002). Inthat review the basic framework of MOND is explained. RCs fits using MOND prior to 2002 are presented and the different capabilities of MOND are listed. The most recent review on MOND and its implications for cosmology can be found in Famaey & McGaugh (2012). The analysis by de Blok & McGaugh (1998) was the first to used low surface brightness (LSB) galaxies in the context of MOND. Those systems are good candidates to test MOND because their accelerations fall below the MOND acceleration limit a 0. de Blok & McGaugh (1998) used a sample of 15 LSB galaxies. They found that MOND is successful to reproduce the shape of the observed RCs for three quarters of the galaxies in the sample. The most recent study of LSB galaxies in the context of MOND was done by Swaters, Sanders & McGaugh (2010). MOND produced acceptable fits for also three quarters of the galaxies in their sample. The correlation between a 0 and the extrapolated central surface brightness of the stellar disc was also investigated. Swaters et al. (2010) found that there might be a weak correlation between a 0 and the R-band central surface brightness of the disc. This is shown in their fig. 7. Their interpretation was that galaxies with lower central surface brightness had lower values for a 0. Such a correlation would be in contradiction with MOND since a 0 should be a universal constant. The reliability of the MOND mass models depends strongly on the sensitivity and spatial resolution of the observed RCs. More extended RCs are needed to trace the matter up to the edge of the galaxies but higher resolution is required in the inner parts. RCs derived from the THINGS sample satisfy these criteria. THINGS consists of 34 dwarfs and spiral galaxies observed in H I with the Very Large Array (VLA) in the B, C and D configurations (Walter et al. 2008). Gentile, Famaey & de Blok (2011) used a subsample of 12 galaxies from de Blok et al. (2008) and modelled their mass distributions using the MOND formalism. They performed one- and two-parameter MOND fits and recalculated the value of a 0 which turned out to be similar to the standard value estimated in Begeman et al. (1991). Their average value for a 0 is cm s 2 using the simple μ-function (Zhao & Famaey 2006) for the interpolation between the Newtonian and the MON- Dian regimes. They also looked at the correlation between a 0 and central surface brightness in the 3.6 μm band and found that there was no correlation. However, their points have a large scatter and they used the bulge central surface brightness instead of the extrapolated disc central surface brightness. Slowly rotating gas-rich galaxies are good candidates to test MOND because their accelerations are below a 0 and the baryonic mass is dominated by gas and not stars. Sánchez-Salcedo, Hidalgo- Gámez & Martínez-García (2013) studied a sample of five such galaxies and found significant departures between the observed RCs and the ones predicted by MOND (see also Frusciante et al. 2012). Recently, Carignan et al. (2013) presented a detailed study of the magellanic-type spiral galaxy NGC 3109 using VLA H I data from Jobin & Carignan (1990) and new data from the Square Kilometre Array (SKA) pathfinder Karoo Array Telescope 7 (KAT-7) telescope in the Karoo desert in South Africa. With both data sets, they found that MOND cannot fit the observed RC of NGC 3109, while their distance and inclination are well determined. On the other hand, the ISO DM halo model gives a very good fit to both data sets. A sample of 15 galaxies, mostly with Cepheid-based distances, will be used for the present study with the aim to minimize the errors coming from the adopted distance. Some RCs have been resampled to have a homogeneous sample of RCs with independent velocity points in order to give significance to the goodness of the fits of the mass models. The mass-to-light ratios (M/L)s used have also been determined in a homogeneous way using stellar population models predictions, instead of leaving them as free parameters. This paper is organized as follows: the sample selection is presented in Section 2, the methods used for the mass models are explained in Section 3, results are shown in Section 4, followed by the discussion in Section 5 and the conclusions in Section 6. 2 SAMPLE The two main selection criteria for the final sample of 15 dwarf and spiral galaxies are the method used to determine their distance and the availability of high-quality H I RCs in terms of spatial resolution and sensitivity. Cepheid-based distances, using their period luminosity relation, are commonly considered as the most accurate for nearby galaxies. We were able to use this method for all but two of the galaxies in the sample. The method used to measure the distance and references are given in column 5 of Table 1. Another criterion is to have a sample of galaxies spanning a wide range of luminosities and morphological types from dwarf irregular to bright spiral galaxies. Therefore, DDO 154 and IC 2574 are included even if Cepheid distances are not available. Furthermore, these two galaxies are gas-dominated galaxies and exhibit large discrepancies between their visible mass and their dynamical mass which makes them ideal objects to test MOND and DM halo models. 11 galaxies of the final sample are part of THINGS, their RCs being derived by de Blok et al. (2008). For the others, the RCs and gas distributions are taken from Carignan et al. (2013) for NGC 3109, from Westmeier, Braun & Koribalski (2011) for NGC 300, from Puche, Carignan & Wainscoat (1991) for NGC 55 and from Carignan & Puche (1990) for NGC 247. There is a considerable overlap between the sample used in this work and Gentile et al. (2011), 7 out of 15 galaxies are common to both studies. However, IC 2574, NGC 925 and NGC 2366 are included in this study, which were omitted by Gentile et al. (2011) because of the presence of holes and shells and non-circular motions. The RCs of these galaxies were derived using the bulk velocity fields only (Oh et al. 2008) which remove the effect of holes and shells. The presence of small bar could also introduce mild distortion, but that is beyond the scope of this study and will be investigated in an upcoming paper which will investigate the non-circular motions induced by the bar as a function of the size and orientation of the bar using numerical simulation.

3 MOND versus dark matter haloes models 3 Table 1. Properties of the galaxies in the sample. Name PA Incl. Distance Method[Ref] m B M B Type ( ) ( ) (Mpc) (mag) (mag) MASS MODELS DDO ± 0.54 bs[k04] IB(s)m IC ± 0.41 rgb[k04] SAB(s)m NGC ± 0.03 cep[g08] SB(s)m NGC ± 0.17 cep[g09] SB(s)cd NGC ± 0.04 cep[g05] SA(s)d NGC ± 0.63 cep[f01] SAB(s)d NGC ± 0.31 cep[t95] IB(s)m NGC ± 0.14 cep[f01] SAB(s)cd NGC ± 1.50 cep[f01] SA(r)b NGC ± 0.25 cep[f01] SA(s)ab NGC ± 0.02 cep[s06] SB(s)m NGC ± 0.95 cep[f01] SB(rs)d NGC ± 0.46 cep[f01] SA(s)d NGC ± 1.02 cep[f01] SA(s)b NGC ± 0.10 cep[p10] SA(s)d Notes. bs: brightest stars, rgb: red giant branch, cep: Cepheid variables; F01: Freedman et al. (2001); S06: Soszyński et al. (2006); K04: Karachentsev et al. (2004); T95: Tolstoy et al. (1995); P10: Pietrzyński et al. (2010); G05: Gieren et al. (2005); G08: Gieren et al. (2008); G09: Gieren et al. (2009). Col. 1: galaxy name; col. 2: position angle; col. 3: inclination; col. 4: distance with the uncertainty; col. 5: method used to measure the distance and reference; col. 6: apparent magnitude taken from the RC3 catalogue (de Vaucouleurs et al. 1991); col. 7: absolute magnitude; col 8: morphology type. 3.1 Mass models with a DM halo As mentioned in the introduction, most of the mass in the galaxies appears to be in the form of unseen matter called dark matter (DM), whose existence is inferred through its gravitational effect on luminous matter such as the flatness of RCs or the gravitational lensing effect. The common scenario is that galaxies are embedded in DM haloes which follow some density distribution profile. In this paper, the observationally motivated pseudo-iso DM halo profile (ISO), characterized by a constant central density core, will be used. A mass model compares the observed RC to the sum of the contributions of the three mass components, namely the gas disc, the stellar disc (and bulge, if present) and the dark halo. It is well known that neutral hydrogen usually has a larger radial extent compared to the luminous part of the galaxy, especially for late-type spirals and dwarf irregular galaxies (at least in the field). Therefore, the RC derived using the H I gas traces the mass of the galaxy to larger radii. The contributions of all the components are required for the mass model. The stellar components used in this study are derived from 3.6 μm surface brightness profiles, converted into mass using the M/L in that particular band. This M/L is assumed to be constant with radius (cf. Section 3.3.2). Most of the gas content of the galaxy is in the form of neutral hydrogen, thus the gas contribution is derived fromthe H I maps corrected for the primordial helium contribution. The DM halo component is derived from the pseudo-iso density distribution. The RC is thus given by the quadratic sum of the contribution from each component: V 2 rot = V 2 gas + V 2 + V 2 halo, (1) where V gas is the gas contribution, V the stars contribution and V halo the contribution of the DM component. The pseudo-iso DM halo model For the pseudo-iso DM halo, the density distribution is given by ρ 0 ρ ISO (R) = 1 + ( R R C ), (2) 2 while the corresponding RC is given by [ V ISO (R) = 4πGρ 0 RC 2 1 R ( )] R atan, (3) R C R C where ρ 0 and R C are the central density and the core radius of the halo, respectively. We can describe the steepness of the inner slope of the mass density profile with a power law ρ r α. In the case of the ISO halo, where the inner density is an almost constant density core, α = Mass models using the MOND formalism MOND was proposed by Milgrom as an alternative to DM. Therefore, in the MOND formalism only the contributions of the gas and of the stellar component are required to explain the observed RCs. In the MOND framework, the gravitational acceleration of a test particle is given by μ(x = g/a 0 )g = g N, (4) where g is the gravitational acceleration, μ(x) is the MOND interpolating function and g N the Newtonian acceleration MOND acceleration constant a 0 As a universal constant, a 0 should be the same for all astrophysical objects. However, observational uncertainties could introduce a large scatter in a 0 and have to be taken into account. Significant departures of a 0 from the standard value could be interpreted as

4 4 T. Randriamampandry and C. Carignan being problematic for MOND. Milgrom estimated a 0 using Freeman s law, which stipulates that disc galaxies have typical extrapolated central surface brightness in the B band (Freeman 1970) of mag arcsec 2. He estimated a 0 to be (0.7 3) 10 8 (M/L) cm s 2. There are other methods which could be used to find a 0, such as the Tully Fisher relation. However, the preferred method, used in many studies, is to estimate a 0 by comparing the computed RCs from mass models to the observed RCs, leaving a 0 as a free parameter. The implications of a 0 in cosmology are explained in Famaey & McGaugh (2012) MOND interpolating functions The shape of the predicted MOND RCs depends on the interpolating function. The standard and simple interpolating functions are mostly used in the literature. The standard μ-function is the original form of the interpolating function proposed by Milgrom (1983a) butzhao & Famaey (2006) found that a simplified form of the interpolating function not only provides good fits to the observed RCs but also the derived M/Ls are more compatible with those obtained from stellar populations synthesis models. The simple μ-function is given as μ(x) = x 1 + x. (5) The MOND RC using the simple interpolating function is given by Vrot 2 = V,b 2 + V,d 2 + V g 2 a 0 r + V,b 2 + V,d 2 + V g 2, (6) where V,d, V,b, V g are the contributions from the stellar disc, the bulge and the gas to the RC. 3.3 Luminous component contribution Gas contribution VLA observations using combined B, C and D array configurations data (Walter et al. 2008) are used to compute the mass density profile of the H I gas for the THINGS galaxies. It is computed using the GIPSY task ELLINT and the tilted ring kinematical parameters from de Blok et al. (2008). The H I profile is corrected by a factor of 1.4 to take into account the Helium and other metals. The output from ELLINT is then used in ROTMOD to calculate the gas contribution in the mass model, assuming an infinitely thin disc. The gas density profiles are taken from Westmeier et al. (2011) for NGC 300 using Australia Telescope Compact Array (ATCA) data and from Carignan et al. (2013) for NGC 3109 using KAT-7 data, from Puche et al. (1991) for NGC 55 and from Carignan & Puche (1990) for NGC 247, using VLA data Stellar contribution For the mass models, the 3.6 μm surface brightness profiles are used for the stellar contribution. 3.6 μm probes most of the emission from the old stellar disc population (Verheijen 1997). It is also less affected by dust and therefore represents the bulk of the stellar mass. The profiles from de Blok et al. (2008) are used for the galaxies from the THINGS sample. The profiles were decomposed into two components for the galaxies with a prominent central bulge (see de Blok et al for more details). The 3.6 μm surface brightness profiles of NGC 55 and NGC 247 are derived in this work. The images were retrieved from the Spitzer Heritage Archive using a 0.6 arcsec pixel 1 scale. After removing the foreground stars, the images were fitted with concentric ellipses using the ELLIPSE task in IRAF. These profiles are corrected for inclination before being converted into mass density. The method from Oh et al. (2008) is adopted to convert the luminosity profiles into mass density profiles for all the galaxies in the sample except for NGC 300 and NGC Oh et al. (2008) first convert the surface brightness profiles in mag arcsec 2 into luminosity density profiles in units of L pc 2 and then convert to mass density using the following expression: [M pc 2 ] = (M/L) (μ 3.6 C 3.6), (7) where (M/L) 3.6 is the stellar M/L in the 3.6 μm band, μ 3.6 the surface brightness profile and C 3.6 is a constant used for the conversion from mag arcsec 2 to L pc 2. The details to find C 3.6 are presented in Oh et al. (2008): C 3.6 = M , (8) where M 3.6 is the absolute magnitude of the Sun in the 3.6 μm band. Using the distance modulus formula and the distance to the Sun, Oh et al. (2008) found = m = (9) The mass density profile of NGC 300 is taken from Westmeier et al. (2011). For NGC 3109, the I-band profile is used instead of the 3.6 μm because it has a larger radial extent. M Mass-to-light ratio (M/L) The conversion of light into mass through the M/L is the principal source of uncertainty in the mass model. Therefore, the determination of this parameter should not be taken lightly or left as a free parameter. In this work, the stellar synthesis models of Bell & de Jong (2001) are used. As mentioned by de Blok et al. (2008), the uncertainties on M/L decrease in the infrared band. Therefore, the M/L derived in the infrared band will be used when available. The method of Oh et al. (2008) is followed. The M/L at 3.6 μm is given by log(m/l k ) = 1.46(J K) 1.38 (10) and (M/L) 3.6 = 0.92(M/L) k (11) The J K colours are those from the 2MASS Large Galaxy Atlas (Jarrett et al. 2003). An I-band M/L of 0.7 predicted by the stellar population synthesis models is adopted for NGC RESULTS The H I RCs of the THINGS sample are oversampled with two data points per resolution element. If we want the reduced χ 2 values to be meaningful, the RCs have to be resampled with only one point per beam, so that every point is independent. Therefore, resampled versions of the THINGS H I RCs and of four other RCs from the literature will be used to construct the mass models of the galaxies in our sample. The gas and stellar contributions to the RCs are computed with the GIPSY task ROTMOD. The outputs from ROTMOD are used in ROTMAS, which is the main task for the mass models. ROTMAS uses a non-linear least-squares method and compares the observed RCs to the calculated RCs derived from the observed mass distribution of the gas and stars (van der Hulst et al. 1992). Inverse squared weighting of the RCs data points with their uncertainties are used during the fitting procedure.

5 4.1 ISO DM haloes fit results DM mass models are shown in the left-hand panels of Fig. 1. The DM halo components are shown as dashed magenta lines, the contribution from the stellar disc components as dot dashed black lines, the stellar bulge components as long dashed green lines and that from the gas components as dashed red lines. The results are summarized in Table MOND results MOND fits with distance fixed The MOND models were performed with the same M/Ls as the ones used for the DM models. MOND fits are shown in the middle panels of Fig. 1 for a 0 fixed at its standard value and in the right panels for a 0 free. The observed RCs are shown as black points with error bars and the MOND RCs calculated from the observed mass density distribution of the stars and gas in continuous blue lines. The stellar disc and bulge contributions are shown as black dot dashed and long dashed lines, respectively, and the gas contributions as red dashed lines. Results for the a 0 fixed and a 0 freefitsarealsoshown in Table 2. An average value for a 0 of (1.13 ± 0.50) 10 8 cm s 2 is found using the simple interpolating function. As for Bottema et al. (2002), this is smaller than the standard value of Begeman et al. (1991) MOND fits with distance let free to vary within the uncertainties. The distance was allowed to vary within the uncertainties in the case of galaxies in which MOND produces poor fits to their RCs. The results are shown in Table 3. 5 DISCUSSION The average reduced chi-squared for the ISO halo(<χr 2> = 1.18) is lower than that obtained from MOND with fixed a 0 (<χr 2> = 9.20) using the distance listed in Table 1 andevenwhena 0 is allowed to vary (<χr 2 > = 2.37). Discrepancies between the RCs predicted by MOND and the observed RCs are seen for most of the galaxies in the sample and particularly for DDO 154, IC 2574, NGC 925, NGC 2841, NGC 3109, NGC 3198 and NGC 7793, in which MOND overestimates or underestimates the rotation velocities. The difference between the MOND fits and the observed RCs is smaller for the following galaxies: NGC 0055, NGC 2366, NGC 3621 and NGC 7331 when a 0 is fixed to its canonical value. Notes on the individual galaxies are given in the appendix, where the results from this study are compared to those in the literature. 5.1 Effect of varying the adopted distances within the uncertainties The results are summarized in Table 3. The adopted distances are different with Gentile et al. (2011) for the following galaxies: DDO 154 (the error bars are not the same), NGC 2403, NGC 3198 (the error bars are not the same) and NGC MOND prefers lower distances for most of the galaxies except for NGC 247, NGC 2403, NGC 2841 and NGC It is worth noticing that a much lower distance is needed for some galaxy such as NGC 3198 if the M/L is fixed. The quality of the fits improves when the MOND acceleration constant a 0 is taken as a free parameter for all the galaxies. MOND versus dark matter haloes models Comparison with previous work There are a wealth of studies on MOND in the literature, but this section will focus on a comparison between this work and that of Gentile et al. (2011) because, not only Gentile et al. (2011) isthe most recent study, but both samples have a large overlap. The main difference between this work and Gentile et al. (2011) isthatin this work, MOND produces acceptable fits for 60 per cent of the galaxies in the sample, while Gentile et al. (2011) found that MOND produces excellent fits for all the galaxies except for three (DDO 154, NGC 2841 and NGC 3198), meaning for 75 per cent of their sample. The similarity between this paper and Gentile et al. (2011) is that seven galaxies are common in both studies. For those seven galaxies, five galaxies produce acceptable MOND fits when a 0 was left free to vary. The differences stem from two main reasons: (i) First, Gentile et al. (2011) allowed the M/L to vary and the distances were constrained within the errors, which gave smaller chi-squared values. In this work, the M/Ls are fixed using those found using population synthesis models. (ii) Secondly, the distance used in this work and Gentile et al. (2011) are not the same, for example the distance adopted in this work for NGC 2403 is (3.22 ± 0.14) Mpc, as listed in Table 1 while their distance is (3.47 ± 0.29) Mpc. Our distances are mainly Cepheid based. However, this work would reach the same conclusions as Gentile et al. (2011) if the M/L was also a free parameter and the distances only constrained. It is well known that the more you have free parameters, the easier it is to get good fits. This is why Gentile et al. (2011) get 75 per cent (9/12) good fits with their distance constrained option and free M/Ls, while this falls to 60 per cent (9/15) when the distance and M/L are fixed. The difference between the two studies is probably not significant given the small sample sizes in both studies. 5.3 Correlation between the MOND acceleration constant a 0 and other galaxy parameters Any systematic trend of a 0 with some galaxy parameter could be a problem for MOND since it is supposed to be a universal constant. Here, we look for a correlation between the corrected central surface brightness of the stellar disc with the MOND parameter a 0.As shown in Fig. 2, galaxies with higher central surface brightness require higher values of a 0 and galaxies with lower central surface brightness lower a 0 values. This has also been seen in the R band for LSB galaxies by Swaters et al. (2010). Gentile et al. (2011) did the same analysis using the 3.6 μm band for twelve (12) galaxies from the THINGS sample and did not find any correlation. However, the bulge central surface brightnesses were used for their study instead of the disc values. Five galaxies in their sample (see their fig. 2) have central surface brightnesses brighter than 13 mag arcsec 2 which are probably due to the small but bright central bulges. The surface brightness profile increases sharply within a small radius and lead to a very high estimated central surface brightness, which is the reason why the extrapolated disc central surface brightness have to be used. The following relation is found in this work which is shown in Fig. 2 as a dashed line log(a 0 ) = ( ± 0.037) μ (4.520 ± 0.687). (12) This surely challenges the universality of MOND, since a 0 should be the same for every galaxy.

6 6 T. Randriamampandry and C. Carignan Figure 1. Mass model fit results, left-hand panel: ISO RCs fits (the parameter results are in Table 2), middle panel: MOND RCs fits with a 0 fixed and right-hand panel: MOND RCs fits with a 0 free (the MOND parameter results are in Table 3). The red dashed curve is for the H I disc, the dash dotted black curve is for the stellar disc, dashed green curves for the stellar bulge and the dashed magenta curves for the DM component. The bold blues lines are the best fit models and the black points the observed rotational velocities.

7 MOND versus dark matter haloes models 7 Figure 1 continued 5.4 DM halo scaling laws The most common hypothesis to explain the flatness of galaxies RCs is to postulate the existence of a DM halo. The halo is characterized by a theoretical density profile. It has been known that the observational motivated ISO halo provides a better description of the observed RCs as compared to the cosmological motivated NFW halo (see e.g. de Blok, McGaugh & Rubin 2001). The success of

8 8 T. Randriamampandry and C. Carignan Figure 1 continued the ISO halo for fitting galaxy RCs has been said to be due to the number of parameters involved in the fitting procedure. These parameters are the halo core radius R C and the halo central density ρ 0. A correlation between these two parameters have been investigated in the literature (Barnes, Sellwood & Kosowsky 2004; Kormendy & Freeman 2004; Spano et al. 2008). Kormendy & Freeman (2004) found the following relation: log ρ 0 = 1.04 log R C 1.02 (13)

9 MOND versus dark matter haloes models 9 Figure 1 continued while Spano et al. (2008) found log ρ 0 = 0.93 log R C (14) A similar analysis is undertaken for our sample. A plot of the core radius as a function of the central densities is shown on the top panel of Fig. 3. We found that our result is consistent with those presented in the literature. This confirms the existence of a scaling relation for the central core of the DM haloes. The following relationship was found using a simple least-squares method: log ρ 0 = ( 1.10 ± 0.13) log R C (1.05 ± 0.25). (15) The core radius is plotted as a function of absolute magnitude in Fig. 4. A clear correlation is seen between these two parameters: low-luminosity galaxies correspond to a small core radius. AsshowninFig.4 our least-squares fit results are log ρ 0 = (0.121 ± 0.021) M B + (1.531 ± 0.120) (16) log R C = ( ± 0.044) M B (1.76 ± 0.797). (17) The least-squares result found by Spano et al. (2008) were log ρ 0 = M B (18) log R C = M B 2.47, (19) and the results by Kormendy & Freeman (2004) were log ρ 0 = M B (20) log R C = M B (21) The correlation between the central density and the absolute magnitude was also investigated which is shown in bottom panel of Fig. 3, this is in good agreement with Kormendy & Freeman (2004).

10 10 T. Randriamampandry and C. Carignan Table 2. Results for the ISO DM halo models with fixed M/L (Diet Salpeter IMF) and for the MOND models using a 0 = cm s 2 and a 0 as a free parameter (simple interpolating function). ISO MOND Name (M/L) 3.6,disc (M/L) 3.6,bulge R C ρ 0 χr 2 χr 2 (a 0 fixed) a 0 χr 2 (a 0 free) (kpc) (10 3 M pc 3 ) (10 8 cm s 2 ) DD [dB08] 1.34 ± ± ± IC [dB08] 7.36 ± ± ± NGC [Tw] 3.17 ± ± ± NGC [Tw] 1.63 ± ± ± NGC [W11] 1.08 ± ± ± NGC [dB08] ± ± ± NGC [dB08] 1.28 ± ± ± NGC [dB08] 4.53 ± ± ± NGC [dB08] ± ± ± NGC [dB08] ± ± ± NGC a [Tw] 2.22 ± ± ± NGC [dB08] 4.85 ± ± ± NGC [dB08] 5.56 ± ± ± NGC [dB08] ± ± ± NGC [dB08] 1.90 ± ± ± <1.18> <9.20> <1.13 ± 0.50> <2.37> a the I-band surface brightness profile was adopted for NGC 3109 because it has larger radial extent than the 3.6 µm surface brightness profile. Col. 1: Galaxy name; col. 2 and 3: mass-to-light ratio [reference: db08: de Blok et al. (2008); Tw: this work; W11: Westmeier et al. (2011); col.4:iso halo core radius; col. 5: ISO halo central density; col. 6: ISO reduced chi-squared. Col. 7 and 9: reduced chi-squared for a 0 fixed and free; col. 8: MOND acceleration parameter. Table 3. Effect of varying the adopted distance within the uncertainties (M/L fixed (Diet Salpeter IMF, a 0 = cm s 2 and using the simple interpolating function). Name (M/L) 3.6,disc (M/L) 3.6,bulge Distance χr 2 (Mpc) DD IC NGC NGC NGC NGC [G11] NGC NGC NGC [G11] Col. 1: Galaxy name; col. 2 and 3: mass-to-light ratio; col. 4: adopted distances, G11: distance used by Gentile et al. (2011) (see text for more explanation); col. 5 : reduced chi-squared. Figure 2. MOND parameter as a function of the corrected central surface brightness of the stellar disc in the 3.6 µm band. The halo surface density is given by the product of the core radius and the central density of the halo. Kormendy & Freeman (2004) found that the surface density of the halo is nearly constant as a function of absolute magnitude. This result was confirmed by Spano et al. (2008) which is shown as a green dashed line in Fig. 3. We found a correlation which is in good agreement with those found in the literature. Kormendy & Freeman (2004): ρ 0 R C 100 M pc 2 (22) Spano et al. (2008): ρ 0 R C 150 M pc 2 (23)

11 MOND versus dark matter haloes models 11 DM ISO halo could be characterized by only one parameter since the core radius and the central density of the halo are correlated. With the M/L of the disc now fixed by population synthesis models, DM models are thus left with only one free parameter the DM surface density. Figure 3. Top panel: core radius as a function of central density for the ISO halo. The bold red lines show the best-fitting result found in this study, the dashed green lines are the correlation found by Spano et al. (2008), the dot dashed blue lines are the correlation found by Kormendy & Freeman (2004) and the filled black circle are the results from this work. Bottom panel: halo surface density as a function of B-band absolute magnitude. Figure 4. Top panel: core radius of the ISO halo as a function of absolute magnitude. Bottom panel: central density as a function of absolute magnitude. The bold red lines show the correlation found in this study, the long dashed green lines are the correlation found by Spano et al. (2008), the dot dashed blue lines are the correlation found by Kormendy & Freeman (2004) and the filled black circle are the parameter results from this work. This work: ρ 0 R C 120 M pc 2. (24) These results confirm the scaling laws for DM haloes, which is important for our understanding of the relation between the dark and luminous matter and the characteristics of the DM itself. It clearly confirms that low luminosity galaxies have smaller core radii and higher central densities. Most importantly, it mainly implies that 6 CONCLUSIONS We have presented mass models of fifteen (15) dwarf and spiral galaxies selected from the literature. Their observed RCs were confronted with MOND and the observationally motivated ISO DM halo model. The galaxies in the sample were selected to be homogenous in terms of their measured distances, the sampling of their RCs and the stellar M/L of the stellar contributions. The selected galaxies in the sample also cover a larger range of luminosities and morphological types than previous studies. The models were carried out using the GIPSY software tasks ROT- MOD and ROTMAS. MOND fits with a 0 fixed and free were performed. MOND fits with a 0 free were needed to re-estimate the average value of a 0,toidentifygalaxiesinwhicha 0 exhibits a significant departure from the standard value of a 0 = cm s 2 and to look for any trend between a 0 and the other parameters of the galaxy. The MOND fit results are as follows. (i) An average value of (1.13 ± 0.50) 10 8 cm s 2 was measured for the MOND acceleration constant a 0 which is smaller compared to the standard value of cm s 2 found by Begeman et al. (1991) and should be considered as the standard value since our sample covers a broader range of luminosities and morphological types. (ii) The RCs predicted by MOND tend to be in good agreement with the observed RCs of bright spirals. (iii) The difference between the RCs predicted by MOND and the observed RCs is the largest for the following galaxies: DDO 154, IC 2574, NGC 925, NGC 3109, NGC 3198 and NGC 7793 which, with the exception of NGC 3198, tend to be low-mass systems. (iv) A correlation between a 0 and the extrapolated disc central surface brightness is found. We found that galaxies with higher surface brightnesses require higher values of a 0 and galaxies with lower surface brightness prefer lower a 0. This is problematic for MOND as a new law of physics since a 0 should be a constant independent of any galaxy property. Finally, for the mass models with DM haloes, the ISO halo provides the best fits to the RCs(<χr 2 > = 1.18) compared to MOND with a 0 fixed (<χr 2>= 9.20) or a 0 free to vary (<χr 2> = 2.37). The existence of the scaling relations for the central core of the DM haloes were investigated for the ISO halo model. A correlation between the central density ρ 0 and the core radius R C was confirmed. This correlation implies that the DM halo model could be characterized by only one of the two parameters. ACKNOWLEDGEMENTS We would like to thank Professor Erwin de Blok for sending us the tabulated RCs and the 3.6 μm surface brightness profiles and also to the THINGS team for making their data publicly available. TR s work was supported by a Square Kilometre Array South Africa (SKA SA) National Astrophysics and Space Science Program (NASSP) bursary. CC s work is based upon research supported by the South African Research Chairs Initiative (SARChI) of the Department of Science and Technology (DST), the SKA SA and the National Research Foundation (NRF).

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S., Famaey B., 2006, ApJ, 638, L9 NOTES ON INDIVIDUAL GALAXIES DDO 154 DDO 154 is a gas dominated nearby dwarf galaxy, classified as an irregular barred galaxy or IB(s). A distance of 4.3 Mpc derived from the brightest blue stars by Karachentsev et al. (2004) is adopted for this study. This is consistent with the previous results adopted in the literature (Carignan & Freeman 1988; Carignan & Beaulieu 1989). The first H I observation of this galaxy was done in 1985 November (Carignan & Freeman 1988). DDO 154 is part of the THINGS sample which consists of 34 dwarf and spiral galaxies. The kinematics and RCs of 19 galaxies from the THINGS were derived by de Blok et al. (2008) and these are the highest quality RCs available to date. The inclination and position angle are shown in Table 1. This galaxy has been studied extensively in the context of MOND. The mass model results are shown in the first row of Fig. 1, the left-hand panel is the model with a DM halo, the middle panel is the MOND model with a 0 fixed and the right-hand panel for a 0 free. Milgrom &Braun(1988) consider DDO 154 as an acute test for MOND because its internal acceleration is deeply in the MOND regime. The poor MOND fit for DDO 154 has been interpreted as being due to the uncertainties on the measured distance since no measured Cepheid-based distance is available. The uncertainties on the inclination is also known to be one of the source of the poor-quality MOND fit for DDO 154 due to the unknown thickness of the H I disc. Recently, Angus et al. (2012) usedanewn-body code which solves the modified Poisson s equation and fits galaxy RCs. They performed four parameters (M/L, stellar and gas disc scaleheights and distance) MOND fits for five galaxies from THINGS. They found that an acceptable MOND fit could be obtained for DDO 154 when the gas disc scaleheight is taken as z g = 1.5 kpc and with a larger M/L. In our study, a good MOND fit is found for a small value of a 0 = 0.68 ± 0.02, nearly half the standard value. NGC 55 NGC 55 is a barred spiral SB(s)m galaxy member of the Sculptor group. A Cepheid distance of 1.9 Mpc was measured by Gieren et al. (2008) as part the Araucaria Project. The H I RCs was derived

13 MOND versus dark matter haloes models 13 by Puche & Carignan (1991) from VLA observations. MOND produces an acceptable fit for the RC of this galaxy. NGC 247 NGC 247 is a nearby dwarf galaxy part of the Sculptor group. This galaxy is classified as SB(s)cd. The most recent Cepheid distance for NGC 247 is 3.4 Mpc. This was measured as part of the Araucaria Project Gieren et al. (2009). The RC of NGC 247 was taken from Puche & Carignan (1991). MOND underestimate the rotation velocities in the inner parts of the RCs which leads to a high reduced-chi-squared value for this galaxy. NGC 300 NGC 300 is a well-known spiral galaxy part of the Sculptor Group classified as SA(s)d. This galaxy has been observed at 21 cm wavelength using the 27.4 m twin-element interferometer of the Owen Valley Radio Observatory (Rogstad, Chu & Crutcher 1979) and using the VLA with the D and C configuration (Puche & Carignan 1991). The H I RC in Westmeier et al. (2011) derived from H I observations using the Australian Compact Array telescope is chosen for this analysis because of its is large radial extend compared to the one done by Puche & Carignan (1991) derived from VLA data. This is one of the galaxies that MOND could not produce acceptable fit to the observed RCs with a reduced chi-squared of IC 2574 and NGC 925 Gentile et al. (2011) mentioned the presence of holes and shells in the H I gas distribution of these two galaxies. The existence of large non-circular motions have also been noticed by Oh et al. (2008). Therefore, the poor-quality MOND fits for these two galaxies could not be interpreted as a failure for MOND. However, the RCs we used were derived using the bulk velocity field only Oh et al. (2008), which takes out most of the effects of local non-circular motions. This is well explained in de Blok et al. (2008). However, the presence of a small bar could also introduce large-scale distortion in the inner parts, but this is beyond the scope of this work and will therefore be investigated in an upcoming paper using numerical simulation. NGC 2366 NGC 2366 is a dwarf galaxy member of the M81 group. The RCs derivedbyohetal.(2008) using the bulk velocity field is adopted for this study. MOND produces an acceptable fit to the observed RC of this galaxy. NGC 2403 This galaxy belongs also to the M81 group of galaxies. It is classified as a barred spiral galaxy SAB(s)cd. NGC 2403 is a well-known bright spiral galaxy. It has been extensively used to test MOND. For example NGC 2403 was part of the sample of Begeman et al. (1991) to estimate the value of the MOND acceleration parameter. The most recent RC of NGC 2403 was derived by de Blok et al. (2008). NGC 2403 is also part of the sample of Gentile et al. (2011) in the context of MOND. A Cepheid distance of 3.22 Mpc is adopted for this work (Freedman et al. 2001). A better fit is obtained with a distance of 3.76 Mpc (see Table 3). NGC 2841 NGC 2841 is a bright spiral galaxy in the constellation of Ursa Major. Many authors have noticed that MOND cannot reproduce the observed RC of NGC 2841 (e.g. Begeman et al. 1991). Sanders (1996) even considered NGC 2841 as a possible case to falsify MOND saying that a good MOND fit was only possible with very large distance and an unrealistic disc M/L. In this work, the discrepancy has largely decreased (χr 2 = 0.896) using the Cepheid distance of 14.1 ±1.5 Mpc (Freedman et al. 2001), and a higher value for a 0 (cf. Table 2) which is consistent with Gentile et al. (2011)(seetheir result with distance constrained). NGC 3031 NGC 3031 or M81 is a bright spiral galaxy in the constellation of Ursa Major. NGC 3031 is located at a distance of 3.63 Mpc (Freedman et al. 2001). The existence of non-circular motions is reported by de Blok et al. (2008). For this reason, it is excluded from the Gentile et al. (2011) sample and the observed RC has not yet been confronted with the MOND formalism. Trachternach et al. (2008) quantified the non-circular motions for this galaxy and found that they lie between 3 and 15 km s 1 for an average of 9 km s 1. They also noticed that the outer disc is warped and that there is some disturbance in the velocity field. Allowing a 0 to vary did not improve the quality of the MOND fit. NGC 3109 NGC 3109 is a nearby SB(s)m dwarf galaxy located at a distance of about 1.30 Mpc from us. The first Cepheid distance of NGC 3109 was measured by Gieren et al. (2005) from a total of 19 Cepheid variables. The first HI RC for this galaxy was derived by Jobin & Carignan (1990) from a VLA observation using C and D configurations. Another H I observation were done using the 64 m Parkes telescope in Australia (Barnes & de Blok 2001), but they could not derive the RC because of the lack of spatial resolution. Begeman et al. (1991) noticed that the gas component need to be increased by a factor of 1.67 after a comparison with single-dish observations in the 21 cm wavelength to be in accord with MOND. Recently, Carignan et al. (2013) obtained new H I observations with the KAT-7 (SKA and MeerKAT precursor) in the Karoo desert in South Africa. Since the short baselines and the low system temperature make the telescope very sensitive to large-scale low surface brightness emission, all the H I gas is detected by KAT- 7. Despite having now the proper gas profile, they conclude that NGC 3109 continues to be problematic for MOND. Since it has the proper gas profile, the RC derived by Carignan et al. (2013)isusedin this study. NGC 3109 still exhibits a large discrepancy between the RC predicted by MOND and the observed RC. This disagreement between the MOND RC and the observed RC remained even when a 0 is taken as a free parameter. NGC 3198 This is a grand design spiral galaxy in the constellation of Ursa Major. The Cepheid distance of Mpc of Freedman et al. (2001) is used in this work. NGC 3198 is a well-studied galaxy in the context of MOND. Many authors have shown that MOND cannot predict the RC of this galaxy with the standard value of a 0, unless adjustment is made on the adopted distance. A much lower distance of 8.6 Mpc is needed to reconcile MOND with the observed